Information Matrix

Our first blog in this series of three looked at forecasting log mortality with penalties in one dimension, i.e. forecasting with data for a single age. We now look at the same problem, but in two dimensions. Figure 1 shows our data. We see an irregular surface sitting on top of the age-year plane. Just as in the 1-d case, we see an underlying smooth surface, and it is this surface that we wish both to estimate and to forecast.

There is much to say on the topic of penalty forecasting, so this is the first of three blogs. In this blog we will describe penalty forecasting in one dimension; this will establish the basic ideas. In the second blog we will discuss the case of most interest to actuaries: two-dimensional forecasting. In the final blog we will discuss some of the properties of penalty forecasting in two dimensions.

This blog has two aims: first, to describe how we go about simulation in the Projections Toolkit; second, to emphasize the important role a model has in determining the width of the confidence interval of the forecast.

In a recent posting I looked at the calculation of percentiles and quantiles, which underpin many calculations for ICA and Solvency II. Simply put, an \(\alpha\)-quantile is the value which is not expected to be exceeded \(\alpha\times 100\)% of the time. This value is denoted \(Q_{\alpha}\). Mathematically, for a continuous random variable, \(X\), and a given probability level \(\alpha\) we have:

$$\Pr(X\leq Q_\alpha)=\alpha$$